Open Access

Tannaka-Krein duality for compact quantum group coactions (survey)


The last decade saw an appearance of a series of papers containing a very interesting development of the Tannaka-Krein duality for compact quantum group coactions on $C^*$-algebras. The present survey is intended to present the main ideas and constructions underlying this development.

Key words: Compact quantum group, coaction, Tannaka-Krein duality.

Full Text

Article Information

TitleTannaka-Krein duality for compact quantum group coactions (survey)
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 3, 282-298
MathSciNet   MR3521698
zbMATH 06630274
Milestones  Received 01/02/2015
CopyrightThe Author(s) 2015 (CC BY-SA)

Authors Information

Leonid Vainerman
Universite de Caen, LMNO, Campus II, B.P. 5186, F-14032 Caen Cedex, France

Export article

Save to Mendeley

Citation Example

Leonid Vainerman, Tannaka-Krein duality for compact quantum group coactions (survey), Methods Funct. Anal. Topology 21 (2015), no. 3, 282-298.


@article {MFAT778,
    AUTHOR = {Vainerman, Leonid},
     TITLE = {Tannaka-Krein duality for compact quantum group coactions (survey)},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {3},
     PAGES = {282-298},
      ISSN = {1029-3531},
  MRNUMBER = {MR3521698},
 ZBLNUMBER = {06630274},
       URL = {},


  1. Saad Baaj, Georges Skandalis, $C^ \ast$-alg\`ebres de Hopf et theorie de Kasparov equivariante, $K$-Theory 2 (1989), no. 6, 683-721.  MathSciNet CrossRef
  2. Yu. M. Berezansky, A. A. Kalyuzhnyi, Harmonic analysis in hypercomplex systems, Kluwer Academic Publishers, Dordrecht, 1998.  MathSciNet CrossRef
  3. Julien Bichon, Hopf-Galois objects and cogroupoids, Rev. Un. Mat. Argentina 55 (2014), no. 2, 11-69.  MathSciNet
  4. Julien Bichon, An De Rijdt, Stefaan Vaes, Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006), no. 3, 703-728.  MathSciNet CrossRef
  5. Florin P. Boca, Ergodic actions of compact matrix pseudogroups on $C^ *$-algebras, Ast\erisque (1995), no. 232, 93-109.  MathSciNet
  6. Kenny De Commer, Makoto Yamashita, Tannaka-Krei n duality for compact quantum homogeneous spaces. I. General theory, Theory Appl. Categ. 28 (2013), No. 31, 1099-1138.  MathSciNet
  7. Kenny De Commer, Makoto Yamashita, Tannaka-Krei n duality for compact quantum homogeneous spaces II. Classification of quantum homogeneous spaces for quantum $\rm SU(2)$, J. Reine Angew. Math. 708 (2015), 143-171.  MathSciNet
  8. Sergio Doplicher, John E. Roberts, A new duality theory for compact groups, Invent. Math. 98 (1989), no. 1, 157-218.  MathSciNet CrossRef
  9. An De Rijdt, Nikolas Vander Vennet, Actions of monoidally equivalent compact quantum groups and applications to probabilistic boundaries, Ann. Inst. Fourier (Grenoble) 60 (2010), no. 1, 169-216.  MathSciNet
  10. Michel Enock, Jean-Marie Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, Berlin, 1992.  MathSciNet CrossRef
  11. T. Hayashi, A canonical Tannaka duality for finite semisimple tensor categories, Preprint: math/9904073 [math.QA] (1999).
  12. Andre Joyal, Ross Street, An introduction to Tannaka duality and quantum groups, in: Category theory (Como, 1990), Springer, Berlin, 1991.  MathSciNet CrossRef
  13. M. Krein, A principle of duality for bicompact groups and quadratic block algebras, Doklady Akad. Nauk SSSR (N.S.) 69 (1949), 725-728.  MathSciNet
  14. E. C. Lance, Hilbert $C^ *$-modules, Cambridge University Press, Cambridge, 1995.  MathSciNet CrossRef
  15. Magnus B. Landstad, Ergodic actions of nonabelian compact groups, in: Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), Cambridge Univ. Press, Cambridge, 1992.  MathSciNet
  16. Magnus B. Landstad, Simplicity of crossed products from ergodic actions of compact matrix pseudogroups, Ast\erisque (1995), no. 232, 111-114.  MathSciNet
  17. Saunders MacLane, Categories for the working mathematician, Springer-Verlag, New York-Berlin, 1971.  MathSciNet
  18. Sergey Neshveyev, Duality theory for nonergodic actions, Munster J. Math. 7 (2014), no. 2, 413-437.  MathSciNet
  19. Sergey Neshveyev, Lars Tuset, Compact quantum groups and their representation categories, Soci\'et\'e Math\'ematique de France, Paris, 2013.  MathSciNet
  20. Sergey Neshveyev, Lars Tuset, Hopf algebra equivariant cyclic cohomology, $K$-theory and index formulas, $K$-Theory 31 (2004), no. 4, 357-378.  MathSciNet CrossRef
  21. Sergey Neshveyev, Makoto Yamashita, Categorical duality for Yetter-Drinfeld algebras, Doc. Math. 19 (2014), 1105-1139.  MathSciNet
  22. Ryszard Nest, Christian Voigt, Equivariant Poincare duality for quantum group actions, J. Funct. Anal. 258 (2010), no. 5, 1466-1503.  MathSciNet CrossRef
  23. Victor Ostrik, Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), no. 2, 177-206.  MathSciNet CrossRef
  24. Claudia Pinzari, John E. Roberts, A duality theorem for ergodic actions of compact quantum groups on $C^ *$-algebras, Comm. Math. Phys. 277 (2008), no. 2, 385-421.  MathSciNet CrossRef
  25. P. Podle\s, Quantum spheres, Lett. Math. Phys. 14 (1987), no. 3, 193-202.  MathSciNet CrossRef
  26. Piotr Podle\s, Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $\rm SU(2)$ and $\rm SO(3)$ groups, Comm. Math. Phys. 170 (1995), no. 1, 1-20.  MathSciNet
  27. Pekka Salmi, Compact quantum subgroups and left invariant $C^ *$-subalgebras of locally compact quantum groups, J. Funct. Anal. 261 (2011), no. 1, 1-24.  MathSciNet CrossRef
  28. Peter Schauenburg, Hopf bi-Galois extensions, Comm. Algebra 24 (1996), no. 12, 3797-3825.  MathSciNet CrossRef
  29. Piotr M. So\ltan, Quantum Bohr compactification, Illinois J. Math. 49 (2005), no. 4, 1245-1270.  MathSciNet
  30. T. Tannaka, Uber den Dualitatssatz der nichtkommutativen topologischen Gruppen, Tohoku Math. J. 45 (1938), no. 1, 1-12.
  31. Reiji Tomatsu, A characterization of right coideals of quotient type and its application to classification of Poisson boundaries, Comm. Math. Phys. 275 (2007), no. 1, 271-296.  MathSciNet CrossRef
  32. Reiji Tomatsu, Compact quantum ergodic systems, J. Funct. Anal. 254 (2008), no. 1, 1-83.  MathSciNet CrossRef
  33. V. G. Turaev, Quantum invariants of knots and 3-manifolds, Walter de Gruyter \& Co., Berlin, 1994.  MathSciNet
  34. K.-H. Ulbrich, Fibre functors of finite-dimensional comodules, Manuscripta Math. 65 (1989), no. 1, 39-46.  MathSciNet CrossRef
  35. R. Vergnioux, KK-theorie equivariante et operateur de Julg-Valette pour les groupes quantiques, PhD thesis, Universite Denis Diderot-Paris 7, 2002.
  36. Shuzhou Wang, Ergodic actions of universal quantum groups on operator algebras, Comm. Math. Phys. 203 (1999), no. 2, 481-498.  MathSciNet CrossRef
  37. Antony Wassermann, Ergodic actions of compact groups on operator algebras. I. General theory, Ann. of Math. (2) 130 (1989), no. 2, 273-319.  MathSciNet CrossRef
  38. Antony Wassermann, Ergodic actions of compact groups on operator algebras. II. Classification of full multiplicity ergodic actions, Canad. J. Math. 40 (1988), no. 6, 1482-1527.  MathSciNet CrossRef
  39. Antony Wassermann, Ergodic actions of compact groups on operator algebras. III. Classification for $\rm SU(2)$, Invent. Math. 93 (1988), no. 2, 309-354.  MathSciNet CrossRef
  40. S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), no. 4, 613-665.  MathSciNet
  41. S. L. Woronowicz, Twisted $\rm SU(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117-181.  MathSciNet CrossRef
  42. S. L. Woronowicz, Compact quantum groups, in: Symetries quantiques (Les Houches, 1995), North-Holland, Amsterdam, 1998.  MathSciNet
  43. S. L. Woronowicz, Tannaka-Krei n duality for compact matrix pseudogroups. Twisted $\rm SU(N)$ groups, Invent. Math. 93 (1988), no. 1, 35-76.  MathSciNet CrossRef

All Issues